In science, researchers typically select a subject of study, measure one or several features of the subject, and then attempt to derive information from the collected data. The obtained data can be broadly categorized into two types. The first category consists of continuous data, usually quantitative, capable of taking infinitely many values that transition smoothly from one to another. Examples include the weight of newborn babies, the velocity of a projectile, or the temperature of a certain material.
In contrast, the second category involves discrete data, generally more qualitative, taking specific, distinct values where transitions represent a notable change in the feature. For instance, whether a lamp is on or off, the presence of a certain component in a chemical solution, or the number of eggs a bird has laid in its nest are examples of discrete data.
An intriguing problem arises when attempting to understand how discrete and continuous data from the same subject interact. A classic example of this interaction can be observed in the study of water. Under normal circumstances, water can exist in three states: solid (ice), liquid (water), and gas (vapor). However, when measuring the temperature of water in the comfort of our homes, we find a continuous range between -30 and 200 degrees Celsius. We have come to understand that, under typical conditions near sea level, water is solid below 0 degrees, liquid between 0 and 100 degrees, and a gas above 100 degrees. Notably, within the spectrum of all possible water temperatures, 0 and 100 degrees are special points, representing walls of some sort where abrupt changes occur.
This project is situated in pure mathematics, specifically in the representation theory of Artin algebras. In this context, researchers typically work with discrete objects, such as isomorphism classes of modules or torsion classes in the module category. However, recent developments have revealed that associated with every algebra in a significant family (finite-dimensional algebras over algebraically closed fields) is a geometric object of continuous nature known as the scattering diagram of the algebra. This diagram was hypothesized to encode much of the discrete information of the algebra and its module category. Subsequently, it was demonstrated that a substantial portion of the scattering diagram of an algebra can be constructed using modules satisfying a specific homological condition known as τ-rigid modules.
This project has several objectives. Firstly, it aims to comprehensively understand scattering diagrams of algebras from both homological and representation theoretic perspectives. Secondly, it seeks to leverage the geometric nature of scattering diagrams to establish new connections and strengthen existing ones between representation theory and other mathematical branches such as algebraic geometry and topology. Thirdly, the project aims to lay the groundwork for the development of scattering diagrams in areas of representation theory that have been recently explored. Finally, it intends to utilize the scattering diagram of algebras to obtain new insights into the behavior of module categories that are currently poorly understood.
All the work conducted in this project is interconnected. For simplicity, we will organize it into the four research axes mentioned above.
To thoroughly comprehend the associated scattering diagram of an algebra, we've pursued two approaches. First, we characterized the stability space of combinatorially defined modules, namely string and band modules. This characterization allows us to completely determine the wall-and-chamber structure of the so-called special biserial algebras, bringing us significantly closer to our objective within this family of algebras. Second, we studied the space of chains of torsion classes in abelian categories, particularly in module categories. This topological space, loosely speaking, is dual to the scattering diagram of an algebra. Our main result in this line of research can be stated as follows: the scattering diagram of an algebra has finitely many walls if and only if the space of chains of torsion classes of the algebra is compact.
We've also explored the connection between the representation theory of algebras and algebraic topology and algebraic geometry through their scattering diagrams, or more precisely, their wall-and-chamber structures. Concerning algebraic geometry, we've demonstrated that the deformation theory of the wall-and-chamber structure of a cluster-tilted algebra of Dinkyn type includes as a subvariety the universal cluster algebra of the same type. Independently, a bridge between representation theory and algebraic topology has emerged. There is a conjecture that for every τ-tilting finite algebra A, there exists a space X that is K(G,1), where G is a group determined by the wall-and-chamber structure of A. The space X is the nerve of a category known as the tau-cluster morphism category of A, which we've shown is also determined by the wall-and-chamber structure of A.
In recent years, higher homological algebra has played a central role in representation theory, with significant efforts to generalize classical notions to this new setting. One of our aims is to develop wall-and-chamber structures and scattering diagrams in higher homological algebra. To achieve this, we begin by studying the concept of higher torsion classes and their relationship with classical torsion classes. Initially, we've demonstrated that every d-torsion class inside a d-cluster tilting subcategory of a module category is the intersection of a classical torsion class in the module category with the said d-cluster tilting subcategory. In a second instance, we've provided an internal characterization of d-torsion classes as those subcategories closed by d-extensions and d-quotients.
Concurrently, we've been exploring abelian categories with enough projective objects, a generalization of the category of finitely presented modules over an algebra. In this context, we've introduced and studied the notion of support τ-tilting subcategories. We've shown that every support τ-tilting category generates a functorially finite torsion class. Additionally, we've demonstrated that these categories recover classical τ-tilting theory and some of its generalizations.
Finally, we've utilized the tools developed in the study of scattering diagrams to address internal problems within the representation theory of algebras. Notably, we've shown that an algebra is τ-tilting finite if and only if the dimension of all bricks (a special type of module) is bounded. Moreover, we've demonstrated that in the module category of every τ-tilting infinite special biserial algebra, there exists an infinite family of bricks with the same dimension vector. We've furthered the understanding of stratifying systems in representation theory by relating a certain class of them to the wall-and-chamber structure of the algebra and by constructing the first example of a stratifying system of infinite size.
The work conducted in this project has brought us to the verge of establishing a coherent scattering diagram for every Artin algebra, a substantial achievement in itself. This progress may also have implications for solving conjectures related to skew-symmetrizable cluster algebras, which fall outside the purview of conventional tools for quiver representations. Furthermore, the enhanced understanding of wall-crossing phenomena in representation theory developed in this project could be leveraged in the mid- to long-term to address longstanding conjectures in algebraic geometry, particularly concerning the study of the stability manifold associated to a triangulated category.
We anticipate that the general theory of support τ-tilting subcategories in abelian categories with enough projective modules will find numerous applications, of which we will mention two here. Firstly, in the monoidal categorifications of cluster algebras and/or the categorifications of cluster algebras arising from algebraic geometry, such as the coordinate ring of the Grassmanian or its positroid variety. In these cases, the general theory gives rise to calculations that quickly become intractable. To circumvent these issues, several non-Artinian associative algebras have been proposed, whose finitely presented modules, or at least some of them, should correspond to the clusters in these cluster algebras. We expect that clusters and mutations in these cluster algebras will be explicitly categorified by support τ-tilting subcategories.
The second application of support τ-tilting subcategories is more applied. In recent years, there has been significant interest in topological data analysis (TDA) and its applications to the training of neural networks. It turns out that persistence theory, the central tool in TDA, is equivalent to the study of finitely presented modules over the nonnegative real line—an abelian category with enough projective objects. Hence, a better understanding of support τ-tilting subcategories in this context could lead to applications in informatics. Moreover, the knowledge obtained from this study will be useful in determining an appropriate category that could facilitate a coherent and computable theory of multipersistence.
Our work on the deformation theory of scattering diagrams for cluster-tilted algebras of finite type has opened the door to the development of a commutative algebra that could be associated with every τ-tilting finite algebra, capturing the combinatorial properties of cluster algebras, even if they are not defined combinatorially. We are currently extending our results to cluster-tilted algebras of infinite type.
Within representation theory, the impacts are already evident. Due to the work of this project and that of other mathematicians, we have arguably established the most crucial conjecture in τ-tilting theory. This conjecture posits that in the module category of a tau-tilting infinite algebra, there is an infinite family of bricks having the same dimension vector. As mentioned earlier, we have settled this conjecture for the class of special biserial algebras. It's noteworthy that this conjecture has only been demonstrated for two other families of algebras.